Random sequences within the framework of algorithmic information theory are discussed in this paper. (An excellent introduction to the area was written by Calude [1].) Specifically, the paper studies the set of random sequences from a topological point of view. Such a study is interesting in itself, and has connections with probability theory. The authors prove that there is a natural topology, which is indeed a relativization of the Cantor topology, with respect to which the set of random numbers is topologically large. The paper is of definite interest to researchers working on the fundamentals of the theory of computation.